A lower bound for coherences on theBrown--Peterson spectrum

Algebraic & Geometric Topology 6 (2006) 287–308 287

A lower bound for coherences on theBrown–Peterson spectrum

BIRGIT RICHTER

We provide a lower bound for the coherence of the homotopy commutativity ofthe Brown–Peterson spectrum, BP , at a given prime p and prove that it is at least.2p2 C 2p � 2/–homotopy commutative. We give a proof based on Dyer–Lashofoperations that BP cannot be a Thom spectrum associated to n–fold loop maps toBSF for n D 4 at 2 and n D 2p C4 at odd primes. Other examples where we obtainestimates for coherence are the Johnson–Wilson spectra, localized away from themaximal ideal and unlocalized. We close with a negative result on Morava-K–theory.

55P43; 13D03

1 Introduction

Recently obstruction theory for imposing E1 –structures on homotopy commutativeand associative ring spectra has been successfully applied in some cases of very well-behaved higher chromatic spectra like En (Goerss–Hopkins [12], Richter–Robinson[23]) and 1E.n/ (Baker–Richter [3]) and up to chromatic type one [3].

The two approaches for such an obstruction theory that are available at the moment weredeveloped by Paul Goerss and Mike Hopkins [12] on the one hand and Alan Robinson[24] on the other hand. They use André–Quillen type cohomology theories as thehabitat for possible obstructions which turn out to be equivalent (see Basterra–Richter[5, Theorem 2.6]). The crucial point in all the examples mentioned above is that onecan rely on an étaleness property of the algebra of cooperations in order to make allobstruction groups vanish.

It is an old, still open question raised by Peter May, whether the Brown–Petersonspectrum, BP , is an E1 ring spectrum. There have been serious attempts to provethat BP has an E1 model. In fact, topological André–Quillen homology defined byMaria Basterra in [4] was originally introduced to solve this problem (compare Kriz[16]).

Unlike complex cobordism, MU , or other Thom spectra, BP is not born with an E1 –structure. Therefore trying obstruction theory methods seems to be a canonical attempt.

Published: 26 February 2006 DOI: 10.2140/agt.2006.6.287

http://www.ams.org/msnmain?fn=705&pg1=CODE&op1=OR&s1=55P43,(13D03)

http://dx.doi.org/10.2140/agt.2006.6.287

288 Birgit Richter

But the algebra of cooperations BP�BP , is a polynomial algebra on countably manygenerators, and this will produce a lot of non-trivial André–Quillen type cohomologygroups.

However, the possible obstruction groups for instance in the setting of Robinson’sGamma cohomology only occur in a small range of degrees. The approach of thisnote is to exploit the sparseness of these degrees together with the sparseness of thecoefficients

BP� D Z.p/Œv1; v2; : : :�; with jvi j D 2pi� 2

and the algebra BP�BP to obtain an estimate for the coherence of the homotopycommutativity of BP . We admit that this lower bound might be much too pessimistic,but it seems that there is some interest in even partial results.

Maria Basterra and Mike Mandell announced that BP has at least an E4 –structure, ie,possesses an action by the little 4–cubes operad. By results of Fiedorowicz–Vogt [10]or Basterra–Mandell this implies that the topological Hochschild homology spectrum ofBP , THH .BP /, is at least an E3 –spectrum and the natural map BP ! THH .BP /

is a map of E3 –spectra.

Our approximation result depends on the prime involved.

Theorem 1.1 The Brown–Peterson spectrum BP at a prime p has at least a .2p2 C

2p � 2/–stage structure.

Here, an n–stage structure is a certain filtration step towards an E1 –structure. AlanRobinson introduced this filtration in [24] and we will describe it in Section 3. We willprove Theorem 1.1 in Section 6.2.

The obstruction theory which we will use is the one developed by Alan Robinson [24;25]. We will explain it in detail in sections 2–4.

At the moment we do not have a direct comparison between n–stage structures andstructures over some version of the little cubes operad.

However, n–stage structures are of independent interest because they give rise toDyer–Lashof operations in a certain range of degrees (see Section 7). In the case ofBP one could get homology operations by comparing the Fp –homology of BP withthe one of MU and H Fp (see Bruner–May–McClure–Steinberger [6, page 63]). Weuse Dyer–Lashof operations to show in Theorem 7.2 that BP cannot be the Thomspectrum associated to an n–fold loop map to BSF . Here n is four for the even primeand 2pC4 for any odd prime p . We stress that a stronger result is mentioned in Lewis’thesis [18].

Algebraic & Geometric Topology, Volume 6 (2006)

A lower bound for coherences on BP 289

In other examples the existence of Dyer–Lashof operations might help for instancewith calculations of topological Hochschild homology.

The main object of study of this note is the Brown–Peterson spectrum. The readerwho is primarly interested in this example might skip sections 2–4 and might procededirectly to Section 5. The material presented in sections 2–4 is needed for cases likethe Johnson–Wilson spectra. We discuss this and other examples in Section 8.

Acknowledgements I thank Sverre Lunøe-Nilsen and John Rognes for their interest.They pointed out to me that my partial coherence results should lead to Dyer–Lashofoperations. Thanks to Andy Baker who read earlier versions of this and made importantcomments. I am grateful to John Rognes and the Department of Mathematics in Oslofor their hospitality. Parts of this work were written when I was in Bonn and I would liketo thank Carl-Friedrich Bödigheimer for his constant support during the last years. Theauthor was partially supported by the Strategisk Universitetsprogram i Ren Matematikk(SUPREMA) of the Norwegian Research Council.

2 Some background on Gamma (co)homology

Let k be a (graded) commutative ring with unit, let A be a (graded) commutativek –algebra and let M be a (graded) A–module. In the following tensor products willall be taken with respect to k . Gamma homology of A over k with coefficients in M

is defined in Robinson [24, 2.5] as the homology of the total complex of a bicomplex„�;� which we will now describe.

Let Lie.n/ be the nth term of the operad which codifies Lie algebras over k , ie, Lie.n/

is the free k –module generated by all Lie monomials in variables x1; : : : ; xn suchthat each variable appears exactly once. There is a canonical action of the symmetricgroup on n letters, †n , on Lie.n/ by permuting the variables xi . Let Lie.n/� be thek –linear dual of Lie.n/. Then the bicomplex for Gamma homology in bidegree .r; s/

is defined as

„r;s.AjkI M / D Lie.s C 1/�˝ kŒ†sC1�˝r

˝ A˝.sC1/˝ M:

Here, all undecorated tensor products are taken over the ground ring k . The horizontaldifferential is the differential of the bar construction; the vertical differential is com-plicated and we refer the curious reader to [24, Secion 2] for details. For our purposeit is enough to know that it induces multiplication of the algebra entries, induces anaction of A on M and does something with the permutations in †sC1 to reduce themto elements in †s .


290 Birgit Richter

We emphasize that all these operations preserve the internal degree if A and M aregraded. Therefore it makes sense to define the i th homogeneous part of „�;� and theassociated total complex. Gamma homology for graded commutative algebras thereforepossesses a natural bigrading, where H�q;i.AjkI M / is the qth homology of the i th

homogeneous part of the total complex of „�;�.AjkI M /. Gamma cohomology, whichis defined via the homomorphism complex out of the total complex of „�;�.AjkI A/

into M , inherits the internal grading. Following [24] we denote by H�q;i.AjkI M /

the qth cohomology of the homomorphism complex

HomiA.Tot.„�;�.AjkI A//; M /

whose morphisms lower degree by i .

3 Robinson’s obstruction theory

Consider the product of the topological version of the Barratt–Eccles operad .E†n/n

and Boardman’s tree operad .Tn/n . Here the space of n–trees, Tn , consists of abstracttrees on n C 1 leaves. These leaves are labelled with the numbers 0; : : : ; n where eachlabel appears exactly once. Internal edges get an assigned length 0 < � 6 1. This treespace is set to consist of a point for n 6 2, the only 2–tree is the tree

��

0

@@

1

2There are three different types of 3-trees, namely

��0

@@

1

��

2

@@ 3��0

@@

2

��

1

@@ 3��0

@@

3

��

1

@@ 2with a corolla-shaped tree if the length of the only internal edge is shrunk to zero.

��0

@@

1

��

2

@@ 3For arbitrary n, the space Tn is contractible with the corolla on n C 1 leaves asbasepoint. Composition in the tree operad is given by grafting trees. The newly builtinternal edge in the composed tree is defined to be of length one. The nth part of theoperad which Robinson uses is the product operad

Bn D E†n � Tn:



As E†n is †n –free and contractible and as Tn is contractible, the product operad Bis an E1 –operad. Robinson defines a filtration of this E1 –operad as follows: setB.i/

n WD .E†n/.i/ � Tn where .E†n/.i/ is the i th skeleton of the standard model forE†n . Then define

(1) rnBm WD B.n�m/

m :

Definition 3.1 [24, 5.3] An n–stage structure for an E1 –structure on a spectrumE is a sequence of maps

�mW rnBm Ë†m

E^m�! E

which on their restricted domain of definition satisfy the requirements for an operadaction on E .

Let us make explicit what that amounts to in small filtration degrees. A 2–stagestructure on a spectrum E consists of action maps starting from r2Bm which is

B.2�m/m D .E†m/.2�m/

� Tm:

The .2 � m/–skeleton of E†m is trivial for m > 2, therefore the only requirement isthat we have a map

..E†1/.1/� T1/ Ë†1

E Š E'

�! E

and that E possesses a map

..E†2/.0/� T2/ Ë†2

E^2Š .†2/ Ë†2

E^2�! E:

This is nothing but a multiplication � on E together with its twisted version � ı � if� denotes the generator of †2 . The axioms of an operad action force the map ' to bethe identity of E . Iterates of � and � ı � act on higher smash powers of E , but theydo not have to satisfy any relations.

A 3–stage structure on E comes with three kinds of maps, because non-trivial valuesfor m are 1; 2; 3. The first value does not give anything new. The second step requiresaction maps

..E†2/.1/� T2/ Ë†2

E^2�! E:

The 1–skeleton of E†2 is the 1–circle, giving the homotopy between � and � ı � .

��

� ı �[[


292 Birgit Richter

In addition to that, the value m D 3 brings in the homotopies for associativity via thetrees we described above, because we obtain maps

..E†3/.0/� T3/ Ë†3

E^3Š .†3 � T3/ Ë†3

E^3�! E:

Theorem 3.2 [24, Theorem 5.5] Assume that E is a homotopy commutative andassociative ring spectrum which satisfies

(2) E�.E^m/ Š HomE�.E�E˝m; E�/ for all m > 1:

If E has an .n � 1/–stage structure which can be extended to an n–stage structure thenpossible obstructions to extending this further to an .n C 1/–stage structure live in

H�n;2�n.E�EjE�I E�/:

If in addition H�n;1�n.E�EjE�I E�/ vanishes, then this extension is unique.

We start with a 3–stage structure. If we want to establish an n–stage structure on E ,then we have to show that Gamma cohomology vanishes in bidegrees .`; 2 � `/ for alln � 1 > ` > 3. (This is incorrectly stated in [24, 5.6] but tacitly corrected in [25, 5.8].)

Let us note that BP satisfies the necessary properties to apply Robinson’s obstructiontheory: BP is a homotopy commutative MU –ring spectrum at all primes, Strickland[27, 2.8,2.9], and it satisfies the requirement from (2).

4 Where the obstructions come from

In this section we will explain the geometric origin of Robinson’s obstruction groups[24]. We do not present anything new here, but focus on some details that we will needlater in Section 8.

The obstruction groups in the theorem above arise from a geometric object: one canextend an n–stage structure to an .nC1/–stage structure if certain cohomology classesvanish. An n–stage always gives rise to maps out of parts of rnC1Bm as well, becausewe require that the action maps �m satisfy the axioms of operad actions where they aredefined. Therefore these actions are closed under composition. Robinson calls the partin rnC1Bm that is generated by compositions the boundary of rnC1Bm and denotesit by @rnC1Bm . Let Qm

nC1be the cofibre of the inclusion of @rnC1Bm [ rnBm into



rnC1Bm and consider the following diagram:

:::

��.@rnC1Bm [ rnBm/ Ë†m

E^m

��

// E

rnC1Bm Ë†mE^m

��

44

QmnC1

Ë†mE^m

��

99sssssssssssssssssssssssssss

:::

We obtain a long exact sequence in E–cohomology

: : : ! E0.QmnC1 Ë†m

E^m/ ! E0.rnC1Bm Ë†mE^m/ !

! E0..@rnC1Bm [ r

nBm/ Ë†mE^m/ ! E1.Qm

nC1 Ë†mE^m/ ! : : :

If the first E–cohomology group of QmnC1

Ë†mE^m vanishes, then one can extend

the action map to the .n C 1/st filtration step. If the term E0.QmnC1

Ë†mE^m/ is

trivial as well, then the extension of the n–stage structure to an .nC1/–stage structureis unique. Note, that we have to consider all 2 6 m 6 n C 1, because these are thenon-trivial domains for an .n C 1/–stage action.

If one has a universal coefficient theorem at hand, one can then continue to identifythis cohomology group as HomE�

.E�.QmnC1

Ë†mE^m/; E�/. Note, that up to this

stage homomorphisms are taken with respect to E� and not E�E . Alan Robinsonthen identifies E�.Qm

nC1Ë†m

E^m/ as some tractable algebraic object and induceshomomorphism up to E�E . Furthermore he proves in [24, Proposition 5.4] thatobstruction are always cocycles for a suitable coboundary map, which then gives thefinal identification with Gamma cohomology groups.

However, one could stop at earlier stages before passing to the level of Gamma coho-mology groups. One can try to extract direct information either out of

(3) E1.QmnC1 Ë†m

E^m/


294 Birgit Richter

or, in the presence of a universal coefficient theorem for E , one can investigate inwhich degrees the groups

(4) Hom1E�

.E�.QmnC1 Ë†m

E^m/; E�/

are non-trivial.

In the examples that we will consider, we already know that the spectra under con-sideration possess A1 –structures. Therefore we have universal coefficient spectralsequences available converging to the cohomological term (3).

Let us examine the E–homology of QmnC1

Ë†mE^m further in order to get control

over internal degree shifts.

We use Robinson’s geometric identification of QmnC1

as E†.n�mC1/m =E†

.n�m/m ^

Tm=@Tm . Here the second quotient is the quotient of the cubical tree complex modulothe subcomplex of all fully grown trees – these are decomposable trees, ie, treeswith at least one internal edge having length one. The first quotient is equivalent toW

†n�mC2m

Sn�mC1 whereas the identification in [24] and Whitehouse [28] shows that

the second quotient has the homotopy type ofW

.m�1/! Sm�2 which on the level ofhomology gives rise to the dual of the Lie representation. One copy of †m is swallowedby the Ë†m

. The degree shift caused by spheres adds up to a total of n � 1, so inE–homology we get the term

†n�1Lie.m/�˝ E�Œ†m�˝.n�mC1/

˝ E�E˝m:

Therefore the E� –homomorphisms on these are given as

Hom1E�

.E�.QmnC1 Ë†m

E^m/; E�/

Š Hom2�nE�

.Lie.m/�˝ E�Œ†m�˝.n�mC1/

˝ E�.E^m/; E�/:

5 The case of BP

Potential obstruction classes for an extension to an .n C 1/–stage structure on BP livein bidegree .n; 2 � n/. Here n is the cohomological degree and 2 � n is the internaldegree. The algebra of cooperations BP�BP over BP� is a polynomial ring overBP� ,

BP�BP D BP�Œt1; t2; : : :� with jti j D 2pi� 2:



We are therefore in the situation where we can apply the universal coefficient theoremto see that the obstruction groups arise from

Hom1BP�

.BP�.QmnC1 Ë†m

BP^m/; BP�/

Š Hom2�nBP�

.Lie.m/�˝ BP�Œ†m�˝.n�mC1/

˝ BP�BP˝m; BP�/:

In order to achieve an internal degree of the form 2 � n, we have to look for a mapthat raises degree by n � 2. Everything in sight is concentrated in degrees of the form

i D

NXiD1

�j .2pj� 2/

where the �j are non-negative integers. Any non-trivial map

Lie.m/�˝ BP�Œ†m�˝.n�mC1/

˝ BP�BP˝m! BP�

can only alter the degree again by a degree of the form i DP

�j .2pj � 2/, becauseBP� is concentrated in the same degrees. The minimal such degree bigger than zero is2p � 2.

Obstruction groups occur as Gamma cohomology of bidegree .n; 2 � n/ for n > 3.Therefore the first possibly non-trivial obstruction group could live in H�2p;2�2p ,which could contain an obstruction to extending a 2p–stage structure to a .2p C 1/–stage structure. This amounts to saying that BP has at least a 2p–stage structure.

6 Refined estimates

The previous section dealt with an argument which was merely a degree count. Wewill improve the 2p–estimate by having a detailed look at the actual homological level.We consider the universal coefficient spectral sequence for Gamma cohomology ofBP�BP . Its E2 –term is of the form(5)Ext�;�

BP�BP .H��;�.BP�BP jBP�I BP�BP/; BP�/ ) H��;�.BP�BP jBP�I BP�/:

Remark 6.1 In the spectral sequence (5) the cohomological degree s for our possibleobstruction group H�s;2�s is now spread over the groups

Ext0;sBP�BP .H��;�.BP�BP jBP�I BP�BP/; BP�/;

: : : ; Exts;0BP�BP .H��;�.BP�BP jBP�I BP�BP/; BP�/


296 Birgit Richter

so we have to prove, that all these groups vanish. As we grade homologically theinternal degree j in Exti;j corresponds to a map that lowers degree by j . Forinstance a homomorphism f in Homj

BP�BP .H��;�.BP�BP jBP�I BP�BP/; BP�/

is a sequence of maps

f D .fs/I fsW H�s;�.BP�BP jBP�I BP�BP/ �! BP s�j :

The reformulations in the rest of this section will not preserve the internal gradingof Gamma homology. The following result is an immediate application of Robinson–Whitehouse [26, 6.8(2)].

Lemma 6.2 Gamma homology groups for BP�BP D BP�Œt1; t2; : : :� split intoGamma homology groups for the single pieces BP�Œti �:

(6) H�s;�.BP�BP jBP�I BP�BP/ Š

Mi>1

H�s;�.BP�Œti �jBP�I BP�BP/:

We start with simplifying each summand in the splitting (6).

Lemma 6.3 Gamma homology of BP�Œti � can be expressed as follows:

H�s;�.BP�Œti �jBP�IBP�BP/

Š BP� ˝Z.p/H�s;�.Z.p/Œti �jZ.p/I Z.p// ˝Z.p/

BP�BP :

Proof For Gamma homology there is a flat-base-change result (see [26, 6.8 (1)])which ensures that

H�s;�.BP�Œti �jBP�I BP�BP/ Š BP� ˝Z.p/H�s;�.Z.p/Œti �jZ.p/I BP�BP/:

Gamma homology of a polynomial generator is ignorant of the taken coefficients: theSteenrod splitting of [23, 4.1] identifies H�s;�.Z.p/Œti �jZ.p/I BP�BP/ with Gammahomology with trivial coefficients induced up to BP�BP

H�s;�.Z.p/Œti �jZ.p/I BP�BP/ Š H�s;�.Z.p/Œti �jZ.p/I Z.p// ˝ BP�BP :

In [23] we proved this in the ungraded case. Gamma homology of a graded algebrahas homogeneous components and the above isomorphism has no reason to preservethe internal grading.

Gamma homology of a polynomial algebra in a single variable with coefficients in theground ring was identified in [23, Proposition 3.2].



Lemma 6.4 Summing over all internal degrees, we can identify Gamma homology ofZ.p/Œti � as M

t>0

H�s;t .Z.p/Œti �jZ.p/I Z.p// Š .H Z.p//sH Z:

Proof Using [23] and the identification of Gamma homology with stable homotopygroups of � –modules from Pirashvili–Richter [22] we obtain

H�s;t .Z.p/Œti �jZ.p/I Z.p// Š

��st

s .Lj .Z.p/Œti �jZ.p/I Z.p/// if j .2pi � 2/ D t

0 otherwise:

where Lj .Z.p/Œti �jZ.p/I Z.p// is the j th homogeneous component of an appropriate� –module. Taking all degrees together gives the claim because in [23, Theorem 4.1] thehomotopy group �st

s .L.Z.p/Œti �jZ.p/I Z.p/// has been identified with .H Z.p//sH Z.

In positive degrees .H Z.p//�H Z is just the p–torsion part of H Z�H Z. Kochman

computed this explicitly in [15, page 44].

6.1 Kochman’s description of .H Z.p//�H Z

Kochman provides in [15, Theorem 3.5 (c)] an explicit basis of the p–torsion inH Z�H Z. The result is:

� There is only simple p–torsion.� An explicit basis of .H Z.p//�

H Z over Z=pZ consists of all expressions

P .n1; : : : ; nt /x�e1

1� : : : � x�es

s

where t > 0, t ¤ 1, 0 < n1 < : : : < nt , ei > 0, t Ce1 C : : :Ces > 0 and ei D 0

for i < n1 . Here, the degree of the P .n1; : : : ; nt / is 2.pn1 C : : : C pnt / � t � 1

and the degree of x�i is 2.pi �1/ with the convention that the degree of P ./ D 1

is zero for t D 0.

Important for us are the cases t D 0 and t D 2. For t D 0 the condition ei D 0 fori < n1 is void, therefore elements like x�

e1

1: : : x�

ess arise with at least one ei being

positive. These elements have total degree

(7) degree.x�e1

1� : : : � x�es

s / D

sXiD1

ei.2pi� 2/:

For t D 2 the situation is a little bit more involved because non-trivial factors likeP .n; m/ occur. The element of lowest possible degree in this case is P .1; 2/ with

(8) degree.P .1; 2// D 2p C 2p2� 2 � 1 D 2p2

C 2p � 3:


298 Birgit Richter

6.2 Proof of Theorem 1.1

Obstructions for extending an n–stage to an .n C 1/–stage structure live in H�n;2�n .From our previous arguments we know that .H Z.p//�

H Z corresponds toH��;�.Z.p/Œti �jZ.p/I Z.p// and consists only of simple p–torsion, ie, summands ofZ=pZ as we are working in Z.p/ –modules. For Gamma homology of BP�BP , whichis given by M

i>0

BP� ˝Z.p/H��;�.Z.p/Œti �jZ.p/I Z.p// ˝Z.p/

BP�BP

we can therefore find a free BP�BP –resolution of length one given by direct sums ofshifted copies of BP�BP

p�! BP�BP .

We therefore obtain that it suffices to consider Ext0;� – and Ext1;� –terms in theUniversal Coefficient spectral sequence (5). We know as well, that the internal degreecan only be of the form

PNiD1 �i.2pi � 2/; consequently possible values for n have

to be of the formPN

iD1 �i.2pi � 2/ C 2 with the �i being non-negative integers.

We are only concerned with positive homological degrees, therefore the correspondingGamma homology groups will be torsion. As there are no non-trivial homomorphismfrom BP�BP=p to BP� the groups

Ext0;nBP�BP .

Mi>0


BP�BP ; BP�/

all vanish.

Thus we are looking for the smallest possible degree of a non-vanishing

Ext1;n�1BP�BP .

Mi>0


BP�BP ; BP�/:

As BP� is concentrated in non-negative degrees, a non-trivial map which lowersdegree by n � 1 can come from Gamma homology in homological degree n � 1 CPN 0

kD1 �k.2pk � 2/. Using the constraint for n gotten above this degree can hitnon-trivial groups only for

n � 1 C

N 0XkD1

�k.2pk� 2/ D

NXiD1

�i.2pi� 2/ C 1 C

N 0XkD1

�k.2pk� 2/:



Considering the degrees of Kochman’s generators P .n1; : : : ; nt /x�e1

1� : : : � x�

ess in the

case t D 0 gives the equation

NXiD1

�i.2pi� 2/ C 1 C

N 0XkD1

�k.2pk� 2/ D

MXjD1

ej .2pj� 2/;

so 1 had to be even and we all believe that this is a contradiction.

The next lowest degrees are to be found for t D2. We content ourselves with consideringa possible cohomology class of degree n � 1. Here the corresponding equation ofdegrees that has to be satisfied is

NXiD1

�i.2pi� 2/ C 1 D 2pn

C 2pm� 3 C

MXjD1

ej .2pj� 2/:

The generator P .1; 2/ is of lowest possible degree and turns this requirement into

n � 1 D

NXiD1

�i.2pi� 2/ C 1 D 2p C 2p2

� 3 D 2p � 2 C 2p2� 2 C 1:

Therefore such a homology class could occur for n D 2p2 C 2p � 2.

Remark 6.5 Of course, the bound in Theorem 1.1 is just an estimate. As some of theidentification did not preserve the internal degree, this obstruction group may be zero.Even if it is non-trivial, it does not have to contain an actual obstruction. Many peoplebelieve that BP is in fact an E1 ring spectrum, and our proof does not disprove that.

The vanishing of Gamma cohomology for degree reasons in a certain range has a secondimportant consequence. Even if there is a homotopy commutative multiplication thatobeys higher coherences, it is an issue whether this structure is unique or whether thereare different ways to find homotopies which care for the higher coherences. Note, thatthe MU � –algebra structure of BP� gives BP a unique MU –ring structure [13, 2.21]and this in turn gives rise to a homotopy commutative and associative ring structure onBP , ie, a 3–stage structure for BP . We get the following estimate.

Corollary 6.6 Any 3–stage structure for BP extends uniquely to a .2p � 1/–stagestructure.

Proof Obstruction groups for the unique extension of an n–stage structure to an.n C 1/–stage structure occur as H�n;1�n.BP�BP jBP�I BP�/. The internal degree1 � n has to be of the form �

PMiD1 �i.2pi � 2/, therefore we obtain the constraint


300 Birgit Richter

n DPM

iD1 �i.2pi � 2/ C 1. For the same reasons as before homological degree n

corresponds to an Ext0;� –term and vanishes. So we have to obtain an estimate for thelowest possible .n � 1/st Gamma homology group. Here, the cases for t D 0 turn outto be relevant. The equation for a possible degree is

n � 1 D

MXiD1

�i.2pi� 2/ D

NXiD1

ei.2pi� 2/:

This can occur for e1 D 1 and ei D 0 for i > 0, so we obtain n � 1 D 2p � 2.Consequently H�2p�1;2�2p.BP�BP jBP�I BP�/ might be non-trivial, and hence agiven .2p�1/–stage structure with a fixed .2p�2/–stage structure might be prolongedto a 2p–stage structure, but in probably different ways.

Remark 6.7 Note, that this estimate for uniqueness is not better than the one wewould have achieved by a mere degree counting argument. We will obtain similarresults later for the localized Johnson–Wilson spectra E.n/.

7 Dyer–Lashof operations

An n–stage structure on a spectrum E gives rise to some low-degree Dyer–Lashofoperations. If E were an H1 –spectrum (see [6]) then the i th Dyer–Lashof operationQi is defined as follows: take the standard resolution, traditionally called W in thatcontext, of the cyclic group of order p , Z=pZ, and assume without loss of generalitythat your spectrum comes with a C W –structure which is compatible with the H1 –structure. Let C� denote cellular chain functor for spectra. Then we obtain a map #

as follows.

# W W ˝Z=pZ C�.E/˝p! C�.E†p/˝†p

C�.E/˝pŠ C�.E†p Ë†p

E^p/ ! C�.E/:

In this situation one can define Qi.x/ for x 2 H Fp�.E/ as

Qi.x/ WD #�.ei ˝ x˝p/:

Here ei is the generator of Wi . Note that this just uses the i –skeleton of E†p ; butthis is part of an n–stage structure on E for i 6 n�p . Therefore we get the followingresult.

Proposition 7.1 If E has an n–stage structure with n > p then there are Dyer–Lashofoperations Qi on the Fp –homology of E for i 6 n � p .



The usual Dyer–Lashof operations Qi are then given by regrading as

Qi.x/ D

8̂̂<̂:̂

0 if i < jxj and p D 2

Qi�jxj.x/ if i > jxj and p D 2

0 if 2i < jxj and p > 2

˙Q.2i�jxj/.p�1/.x/ if 2i > jxj and p > 2:

Andy Baker noticed a curious fact about Dyer–Lashof operations on BP . First,let p be an odd prime. The indecomposable element ap�1 2 .H Fp/2p�2.MU / isknown to be in the image of .H Fp/2p�2.BP/. Consider an element x D x2p�2 in.H Fp/2p�2.BP/ with image ap�1 . For such an x the highest Dyer–Lashof operationQi which we get out of the .2p2 C 2p � 2/–stage structure is Q2p . In [14, 2.11]Hu, Kriz and May proved that the inclusion from BP to MU cannot be a map ofcommutative S –algebras, and they used this particular Dyer–Lashof operation to showthat (compare the correction of the proof of [14, 2.11] in [1, Appendix B]). The imageof ap�1 under Q2p is a.2pC1/.p�1/ up to decomposable elements, but there is noindecomposable element in .H Fp/.2pC1/.p�1/.BP/. For p D 2 a similar argumentworks using a1 .

In the following SF denotes the colimit SF D colimSF n where SF n is the monoid ofbased homotopy equivalences of Sn of degree one, so BSF is the classifying space ofspherical fibrations. We emphasize that the following result is not optimal. In his thesisLewis [18, pages 145–6] wrote down a sketch of an argument due to Priddy, that BPcannot be the p–localization of a Thom spectrum associated to a map of H –spacesfrom a double loop space X to BSF . Priddy’s argument involves the Eilenberg–Moorespectral sequence and calculations with secondary cohomology operations.

I thank Stewart Priddy and Yuli Rudyak for telling me that Lewis’ thesis still is the onlywritten account of that argument. We offer our weaker result here because we think thatthe proof via Dyer–Lashof operations which we give here is short and straightforward.

Theorem 7.2 The Brown–Peterson spectrum BP cannot be the p–localization of aThom spectrum associated to a 4–fold loop map to BSF at p D 2 resp. a .2pC4/–foldloop map to BSF at any odd prime p .

Proof Assume there were such a map from an n–fold loop space X to BSF

W X �! BSF

which would allow to write BP as the Thom spectrum associated to , BP D X .Lewis’ result states that Thom spectra associated to n–fold loop maps to BSF are


302 Birgit Richter

En –spectra [19, Theorem IX.7.1]. Here En is the product of the little-n–cubes operadand the linear isometries operad.

The Thom isomorphism tells us that the homology of BP is isomorphic to the homologyof X , and the latter maps to the homology of BSF . The isomorphism respects theDyer–Lashof operations [19, Proposition IX.7.4 (i)]. In the following H will denotehomology with Fp –coefficients.

If p D 2, the homology of BSF is

(9) H�.BSF / Š H�.BSO/ ˝ C�;

whereas at odd primes, the homology of BSF is isomorphic to

H�.BSF / Š H�W ˝ C 0�:

Explicit formulæ for C� and C 0� can be found in [9, page 114]. The map from BSO

to BSF is an infinite loop map and it is this map which includes the tensor factorH�.BSO/ into H�.BSF /. Therefore the tensor factor H�.BSO/ is closed under theDyer–Lashof operations. A similar remark applies to W which is a summand of BO

at odd primes, because there is a splitting of infinite loop spaces

BO.p/ ' W � W ?

for any odd prime p . In particular, for x D x2p�2 in H�.BP/ with P1�.x/ D 1 we

obtain a non-trivial class of degree 2p � 2 in H�.BSF /. From [9, pages 114–5] itis clear that there is no class of that degree in the C� – resp. C 0

� –part of the abovetensor product, and therefore x has to have an image in H�.BSO/ resp. H�.W /.In both cases, x has to hit an indecomposable element, whose Q2p –image gives agenerator up to decomposable elements. The lack of indecomposables in H�.BP/

yields a contradiction.

Example 7.3 One can ask whether our approach yields any new Dyer–Lashof op-erations for the topological Hochschild homology of the Brown–Peterson spectrum.An E3 –structure on THH .BP / justifies the calculation of THH .BP / given in [21,pages 23–4] as

THH .BP /� Š BP� ˝ ƒ.�1; �2; : : :/; degree.�i/ D 2pi� 1:

See [7, Section 6] as well.

A pure degree estimate or even a more detailed analysis than in the proof of Theorem1.1 gives a possible obstruction class in cohomological degree H�2p;2�2p caused by�1 . Therefore THH .BP / has at least a 2p–stage structure for all primes. In terms of



Dyer–Lashof operations this gives Qi.x/ for .2i � jxj/.p � 1/ 6 p , thus 2i � jxj < 2

at odd primes and for .i � jxj/ 6 2 at 2. An E3 –structure would provide operationsQi on x for 2i � jxj < 2 at odd primes and i � jxj < 2 at 2 (compare [6, III.3.1]).

Browder operations

The geometry of n–stage structures gives rise to additional homology operations, whichgive homological obstructions to extending an n–stage structure to an .n C 1/–stagestructure. Recall that an n–stage structure includes actions by operad subspaces whichcome from operadic composition of lower stages. For instance an operad product

W .E†.0/2

� T2/ � .E†.0/i � Ti/ � .E†

.0/j � Tj / ! .E†

.0/iCj�1

� TiCj�1/

gives elements in †nC1 � @TnC1 for i C j � 1 D n C 1 which act on E if the ringspectrum E admits an n–stage structure where @TnC1 � TnC1 is the subspace offully-grown .n C 1/–trees. We already used the identification of the homology of thequotient TnC1=@TnC1 as the dual of the Lie representation. As TnC1 is contractible,the H Fp –homology of @TnC1 is isomorphic to .H Fp/

�C1.TnC1=@TnC1/. On thehomological level we obtain therefore an action of this shifted copy of Lie.n C 1/� :

(10) ‚�W .H Fp/�.†nC1 � @TnC1/ ˝†nC1

.H Fp/�.E/˝nC1

Š .H Fp/�.@TnC1/ ˝ .H Fp/

�.E/˝nC1

�! .H Fp/�.E/:

If the n–stage structure on E could be refined to an .n C 1/–stage structure, then thisoperation has to be trivial, because it factors over the entire tree space TnC1 which iscontractible.

Cohen’s Theorem 12.3 in [9] expresses the action of the top-dimensional class of thek –configurations in Rm , F.Rm; k/, with an iterated Browder operation. The homologyof configuration spaces in turn is related to free Lie algebras. In [9, pages 263–4] resp.[8, Theorem 6.1] Cohen describes the homology of F.Rm; k/. For an even m > 0,

H.k�1/.m�1/.F.Rm; k/I Z/ Š Lie.k/

as a ZŒ†k �–module 1. Up to suspension and a process of dualization, this is preciselythe homology of the space of fully grown .n C 1/–trees, for k D n C 1.

In the context of iterated loop spaces, a non-trivial Browder operation �n (comingfrom an .n C 1/–fold loop space structure on a space) is an obstruction to extending

1Note that Cohen considers Lie.k/ without the sign-representation [8, page 32] whereas Robinsontwists the †k –action on Lie.k/ by the sign [24, page 333]. Therefore we have to adjust Cohen’s statementto the above form.


304 Birgit Richter

this structure to an .n C 2/–fold loop space structure. The geometric operation from(10) relates this to obstructions for .n C 1/–stage structures.

It would be of interest to clarify some of the open questions about possible E1 –structures on spectra by finding non-trivial operations as in (10). We hope that someonewho is more skillful with calculations than the author might actually succeed to findobstruction classes.

8 Other examples

In the following we discuss examples where we obtain estimates for homotopy coher-ence by a mere degree count, either because the actual (co)homology calculation wouldbe hard if not impossible or because we come across a cohomology class for whichwe have good reasons to believe that it is an actual obstruction class. The localizedJohnson–Wilson spectra are examples of the first type whereas the Morava-K–theoriesare examples of the second kind.

8.1 Localized Johnson–Wilson E.i /

We start by considering the localized version of the Johnson–Wilson spectra E.i/, E.i/

with E.i/� Š .BPhii�/Ii(compare [2]). The coefficients of E.i/� are the coefficients

of BPhii localized away from the ideal Ii D .p; v1; : : : ; vi�1/. Therefore units inE.i/� are of the form sv

ji C r with s a unit in the p–local integers, j 2 Z and r 2 Ii

(compare [2, Section 1]). By [2] it is known that E.i/�E.i/ is free over E.i/� . Thuspossible obstructions live in

Hom2�nE.i/�

.Lie.m/�˝ E.i/�Œ†n�mC1

m � ˝ E.i/�E.i/˝m; E.i/�/:

As E.i/ is Landweber exact over BP , the algebra of cooperations can be described as

E.i/�E.i/ D E.i/� ˝BP�BP�BP ˝BP�

E.i/�:

The degrees of this algebra and hence the degrees of the E.i/� –module

Lie.m/�˝ E.i/�Œ†n�mC1

m � ˝ E.i/�E.i/˝m

are concentrated in degrees of the form

(11)MXiD1i¤n

�i.2pi� 2/ C �n.2pn

� 2/

with the �i being non-negative integers and �n 2 Z.



As this is the underlying module for the chain complex whose dual gives Gammacohomology we obtain the following result.

Proposition 8.1 The i th localized Johnson–Wilson spectrum E.i/ possesses at leasta 2p–stage structure which is unique up to the .2p � 1/–stage.

Proof For any potential obstruction we need a non-trivial E.i/� –linear morphismwhich raises degree by some number of the form n � 2 with n > 3. As the domainLie.m/� ˝ E.i/�Œ†n�mC1

m � ˝ E.i/�E.i/˝m and the target E.i/� are concentrated indegrees as calculated in (11) the minimal such difference of degrees is 2p . Thereforethere might be an obstruction to extending a given 2p–stage structure to a .2p C 1/–stage structure.

8.2 Ordinary Johnson–Wilson spectra

The i th Johnson–Wilson spectrum E.i/ is Landweber exact and a homotopy com-mutative ring spectrum for all primes. The algebra of cooperations E.i/�E.i/ is flatover E.i/� and hence the Künneth theorem applies, but we have to use the universalcoefficient spectral sequence to control the source E.i/1.Qm

nC1Ë†m

E.i/^m/ ofpossible obstructions for highly coherent homotopy commutative structures. TheE2 –term of this spectral sequence is as follows:

Extp;q

E.i/�.E.i/�.Qm

nC1 Ë†mE.i/^m/; E.i/�/

Š Extp;q

E.i/�.†n�1Lie.m/�

˝ E.i/�Œ†m�˝.n�mC1/˝ E.i/�E.i/˝m; E.i/�/

Here the .n � 1/st suspension in E.i/–homology comes from the geometry of thefiltration quotient Qm

nC1as it was discussed in Section 4.

In this spectral sequence we just have to consider the 0– and 1–line, becauseE.i/�E.i/ is a countable colimit of free E.i/� –modules and the other tensor factors arefree. As we are only interested in cohomological degree one and Extp;q

E.i/�corresponds

to cohomological degree p C q we have to determine, for which n > 3 there arenon-trivial Ext0;1

E.i/�– and Ext1;0

E.i/�– terms. Taking the internal .n � 1/–shift into

account we have to look out for homomorphisms which raise degree by n � 2 resp.n � 1. Counting degrees once again we obtain the following result.

Proposition 8.2 The i th Johnson–Wilson spectrum E.i/ possesses at least a .2p�1/–stage structure which is unique up to the .2p � 2/–stage.


306 Birgit Richter

Proof Both sides are concentrated in degrees of the form

MXiD1i¤n

�i.2pi� 2/ C �n.2pn

� 2/:

Due to the possibility of a non-trivial Ext1;1�n –term, the lowest possible value for n

is n D 2p � 2 C 1 D 2p � 1.

Remark 8.3 Note that the above estimate is much too weak for i D 1 because E.1/

possesses a unique E1 –structure at all primes [3, Theorem 6.2].

8.3 Morava-K –theoryLast but not least we will close with a negative result. It is known that none of theMorava-K–theories could possess E1 –structures: if K.n/ had one, then its connectivecover k.n/ had one as well; in particular, k.n/ would be an H1 –spectrum whichthen had to split as a wedge of suspensions of Eilenberg–MacLane spectra (see [6, III,Theorem 4.1]). The argument in [6, III, Theorem 4.1] even shows that k.n/ cannotpossess any H2 –structure. From [20, 2.10] it follows that the connected cover functorc [20, VII.3.2] sends H2 –spectra to H2 –spectra. Therefore K.n/ cannot have anE2 –structure, because this would give rise to such an H2 –structure on k.n/.

At p D 2, K.n/ is even not homotopy commutative. For MU.p/ –algebra structures onK.n/ see [11; 17]. At odd primes, the K.n/ are homotopy commutative and associativering spectra.

Consider K.n/ for an odd prime p . We know that the algebra of cooperationsK.n/�.K.n// for K.n/ consists of an étale part tensored with an exterior algebraƒK.n/�

.�0; �1; : : : ; �n�1/ with the �i being elements of degree 2pi �1. This non-étalepart could give rise to possibly non-trivial obstructions. Here �0 is of internal degreeone. A possible non-trivial obstruction groups could therefore be obtained by a classin cohomological degree n D 2 C 1 D 3. Hence even at odd primes there might be anon-trivial obstruction to extending the homotopy commutative structure on K.n/ to a4–stage structure.

Remark 8.4 The situation for the spectrum P .n/ at odd primes with P .n/� D

Fp Œvn; vnC1; : : :� is the same as the one for Morava-K–theory. This spectrum cannotpossess an H2 –structure. The algebra of cooperations is

P .n/�P .n/ D P .n/� ˝BP�BP�BP ˝ ƒ.a0; : : : ; an�1/

with ai being of degree 2pi � 1; in particular a0 has degree one, which should giverise to an obstruction to a 4–stage structure.



In both cases, K.n/ and P .n/, there are candidates for non-trivial Browder operationsfor K.n/–homology respectively H Fp –homology, but we have no proof so far thatthese homology classes actually arise in this way.

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Fachbereich Mathematik der Universität Hamburg, Bundesstraße 5520146 Hamburg, Germany

[email protected]

http://www.math.uni-hamburg.de/home/richter/

Received: 25 May 2005 Revised: 17 November 2005



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A lower bound for coherences on theBrown--Peterson spectrum

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